AP Photo/Nabil al-Jurani
Trading house Castleton Commodities International has offered $1 billion for it, according to the Financial Times, which cites a person familiar with the matter.
Castleton is backed by hedge fund manager Paul Tudor Jones and Glenn Dubin.
Russian state-owned oil company Rosneft had tried to Morgan Stanley's oil trading unit, but abandoned the deal in December because of US sanctions on Russia over the Ukraine crisis.
The sale would include contracts and inventories, the FT reported.
As oil and other commodity prices continue to slide, Wall Street banks have been scaling back trading. And following the financial crisis, new regulations ban the banks from owning global commodities businesses.
Earlier this month, Morgan Stanley agreed to spin off its natural gas business to a newly formed company called Pentagon Energy LLC.
Selling the oil business this year is a priority for Morgan Stanley CEO James Gorman, according to the Wall Street Journal.